As StoneyB points out, that is acting as a relative pronoun here, giving a sense like “heath that the wind whirls away” or “heath which the wind whirls away”.
As FumbleFingers points out, the two lines are from a poem: Fallen is Thy Throne by Thomas Moore
(born 1779, 36 years before Charlotte Brontë).
I don't know the original publication date of Fallen is Thy Throne, except that apparently its first wide availability was in Moore's Sacred Songs [1] in May or June, 1816. Thus the “wild wind whirls away” quotation may be an anachronism: I've supposed the time frame of Jane Eyre to be ca. 1809 or 1810, based on reference in Chapter XXXII to Scott's Marmion (late 1808) as “a new publication”:
“... I have brought you a book for evening solace,” and he laid on the table a new publication—a poem ...
While I was eagerly glancing at the bright pages of “Marmion” (for “Marmion” it was), St. John stooped to examine my drawing. ...
[1] A Series of Sacred Songs, Duetts and Trios, The Words by Thomas Moore, Esqr. The Music, Composed and Selected by Sir John Stevenson, part 1 (London: J. Power / Dublin: William Power, 1816; Philadelphia: Published by Geo. E. Blake, 1817?); part 2 (London: J. Power, 1824).